How to export physics data from Blender 2.80 now that BGE is gone?
Looks like I can answer my own question.

After some digging I found another simple add-on compatible with Blender 2.8 (this time called "bullet-blender-exporter", LOL) that exports a .json instead of a serialized .bullet file. A nice work-around for anyone that needs to export quickly the rigid bodies and constraints from Blender.

For anyone interested the add-on is this: https://github.com/xu-xionglong/bullet-blender-exporter